Reducing the Standstill Spacing and Duration of Safe Braking Control in Vehicle Platoons With Delays

Braking control, especially in an emergency, is a key technology that is needed to ensure safety in vehicle platoons. However, delays in vehicle platoons can severely affect braking control. This paper proposes an optimized braking control to reduce the standstill spacing and braking duration during delays so that a platoon will stop within a short time frame with a reduced length, thus improving road utilization while ensuring both inter-vehicle and in-vehicle safety. However, two challenges need to be addressed. First, due to the delay in car-following interactions and the nonlinearity of the control law, an analysis model is needed to quantize the duration and distance during an emergency braking with delays. Second, the optimization of these control parameters is an NP-hard problem. Therefore, delay differential equations are introduced to model the braking process, and a crossing criterion is introduced to establish the relationship between the control law and the braking process. The propositions of standstill spacing and braking duration are then derived based on the Runge-Kutta method. According to these criteria, a particle swarm optimization (PSO) based on a lexicographic method with a penalty function is introduced to provide a solution framework with polynomial complexity. Simulation results verify the accuracy of the braking modeling process. Moreover, the results verify the performance of the proposed algorithm and provide a reference for platoon control.


I. INTRODUCTION
As an important component of intelligent transport systems (ITSs), platoon-based driving patterns have attracted significant attention and are characterized by advantages including increased road safety, improved road capacity, and the reduction of energy consumption [1]. Many communities are aimed at furthering this work, such as the SARTRE program, PATH research, the Grand Cooperative Driving Challenge (GCDC), Energy ITS, and SCANIA platooning [2].
Braking control, especially in an emergency, is a key technology in vehicle platoons and is critical to ensuring both inter-vehicle and in-vehicle safety. Regarding the former, to avoid rear-end collisions, vehicles must be kept at a safe distance from each other during braking. Regarding the The associate editor coordinating the review of this manuscript and approving it for publication was Yan-Jun Liu. latter, to avoid an overloaded braking capacity and injuries to passengers [3]- [5], deceleration must be limited under the permissible maximum value.
There are various types of delays existing in practical platoon systems, which can be divided into two categories according to the execution stages of the braking control process: signal acquisition delays and control parameter calculation delays. The first category mainly includes communication delays [6] and sensor delays [7], [8]. Communication delays are caused by information transmission and the queuing of data packets, which are restricted by wireless channel quality and spectrum bandwidth; this type of delay ranges from a few milliseconds to tens of milliseconds. Sensor delays are caused by measurement, data sampling, obstacle detection, ego-positioning, environmental perception, and classification factors. For vision measurement systems in particular, image processing also takes time. This type of delay VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ can range from tens of milliseconds to hundreds of milliseconds. Delays due to the calculation of control parameters can range from a few milliseconds to hundreds of milliseconds.
Delays have a negative impact on vehicle driving control. For an individual vehicle, delayed information can impair its ability to maintain a desired velocity and position [9]. For a platoon, without a proper driving control design, the delayed information will spread from the leader to the followers, threatening stability and creating deviations from the desired control target [10]. For emergency braking control in particular, the urgent timing and reduced spacing can lead to a greater risk of collision, which calls for a more rigorous design to overcome delays.
To ensure braking safety for platoons experiencing delays, it is beneficial to increase the inter-vehicle distance and decrease the velocity, as these measures can provide adequate spacing and time. However, these adjustments will reduce road utilization. Therefore, this paper proposes an optimized braking control to reduce standstill spacing and braking duration during delays while guaranteeing safety. The platoon must stop within a short time frame when an emergency occurs, and the length of the platoon must be shortened after the platoon stops to improve road utilization. However, the distances between vehicles must be greater than a safe distance during the braking process to avoid collisions, and the vehicle decelerations must be less than the maximum permissible value to minimize the damage to cars and the impact on passengers, thus ensuring both inter-vehicle and in-vehicle safety. However, two challenges need to be addressed. First, due to the delay in car-following interactions and the nonlinearity of the control law, an analysis model is needed to quantize the duration and the decreasing distance during the elapsed braking control time frame with delays. Second, the optimization of these control parameters is an NP-hard problem. To address these challenges, the standstill spacing and duration are theoretically analyzed by delay differential equations (DDEs), and a multiobjective and multivariable particle swarm optimization (PSO) algorithm is designed.
The main contributions of this paper are summarized as follows: 1) An analytical expression of the braking process is first modeled using DDEs to embed the effect of delays; 2) A crossing criterion is introduced to establish the relationship between the control law and the braking process and to classify various braking scenarios; 3) The propositions of standstill spacing and braking duration are derived based on the Runge-Kutta method; 4) When considering delays, the standstill spacing and braking duration of the platoon are shortened during emergency braking to improve road utilization while ensuring both inter-vehicle and in-vehicle safety, and a solution framework with polynomial complexity is provided.
The remainder of this paper is organized as follows. Section II introduces related research. Section III presents the system model and the optimization problem. Section IV analyzes the standstill spacing and duration via the DDE method. Section V proposes a multiobjective and multivariable PSO to solve the optimization problem. Section VI reports the results of simulations to verify the performance improvement of the proposed algorithm. Finally, Section VII concludes this paper.

II. RELATED WORKS
Delays in vehicle platoons have recently attracted substantial research attention. Various control policies have been designed for diverse objectives under certain delays, such as the achievement of consensus [11], the robustness for formation changes and control [12], [13], the consensus of vehicles and their consistency with traffic flow theory [14], coping with the curves and slopes of roads [15], [16], and spacing [17]. There have also been several studies concerning the combination of control and delay bounds under stable conditions [6], [18]- [20].
As a key component of platoon management, braking can be classified into two categories: target stopping position (TSP) and emergency braking. Regarding the first category, Li et al. proposed an integral sliding-mode control, and Liu et al. utilized the force approach to achieve cooperative TSP braking control [21]- [23]; however, the restriction of deceleration was only considered in one of these works [21]. Although delay was not considered in these TSP problems, the requirements induced by the attainable deceleration and the negative effect of delays could be resolved by increasing the braking distance because the stopping position was given before braking. However, regarding the second category of braking, namely, emergency braking, delays will greatly increase urgency. In a previous study, an experimental platform and an actual-vehicle experiment were set up to evaluate emergency braking [24], but delay was not considered. Several works have considered delays in emergency braking [25]- [27]. For instance, Flores et al. [25] presented a cooperative collision avoidance system based on pedestrian prediction, in which delay was considered merely in the correction of velocity, and in-vehicle safety was ignored. Xu et al. [26] compared different communication information structures and contents to analyze the intrinsic relationships between coordination and communication in platoons and enhance platoon safety, including during slow and fast braking. Thunberg et al. [27] analyzed the platoon dynamics during emergency braking via packet loss probabilities and packet transmission delays. While the limitation of deceleration to guarantee in-vehicle safety was considered in these works, the effect of delays on braking was solely reflected in deriving the delay bounds to ensure safety [26], [27]. Unlike in the existing literature, in this paper, the control parameters are optimized to improve road utilization while guaranteeing safety, and delays in car-following interactions during braking control are considered.

III. SYSTEM MODEL A. PLATOON DESCRIPTION
As illustrated in Fig. 1, this paper considers a homogeneous platoon that includes a platoon leader (PL) and N platoon members (PMs). The length of each vehicle is denoted as L, the time variable is denoted as t, and the position of the ith vehicle at time t is denoted as x i (t), where i = 0 is the PL and i = {1, 2, . . . , N } represents the PMs. It should be noted that the position of each vehicle is set as its center; thus, the distance between the centers of the (i-1)-th vehicle and ith vehicle at time t, denoted as d i−1,i (t), can be obtained as In this paper, the information topology of platoon control is considered to be a predecessor-follower topology, in which each vehicle obtains the information from its predecessor, including the predecessor's velocity and position. The delay from the (i − 1)-th vehicle to the i-th vehicle is denoted as τ .
Here, delay τ is assumed to be time-invariant in this paper.

B. CONTROL MODEL
The optimal velocity model (OVM) is used in this paper, in which the acceleration or deceleration of the vehicle is determined by two aspects of the velocity difference, namely, the difference between the headway-dependent velocity and the actual velocity and the difference between the velocity of a given vehicle and the preceding vehicle [6], [28]. As a result, the control input (acceleration or deceleration) of the i-th PM, which is denoted as u i (t), is determined by the following control scheme [29]: where a and b denote the associated gains of the aforementioned two aspects of the velocity difference. In addition, the headway-dependent velocity V (d) is determined as follows [6], [28]: In a dense scenario, the vehicle will stop for the distance d < d dense . In contrast, in a sparse scenario, the vehicle will travel with the maximum velocity, denoted as v max , when d > d sparse . For d dense ≤ d ≤ d sparse , the desired velocity will increase linearly with d. The expression of u i (t) is nonlinear due to the nonlinearity of V (d).
Assume that the maximum velocity v max is a constant, and the original state of the platoon before braking is stable, i.e., u i (t) = 0, v i (t) = v stable , and d i−1,i (t) = d stable . The relationship between d stable and v stable can be derived from Eqs. (1) and (2) as follows: From an analysis of Eqs. (1) and (2), it can be found that, even for the same delay and the aforementioned two aspects of the velocity difference, different control parameters, including a, b, d dense and d sparse , will result in different braking processes. Hence, the control parameters must be optimized to improve road utilization while ensuring safety.

C. OPTIMIZATION PROBLEM STATEMENT
Consider a scenario in which the PL detects an emergency directly in front of the platoon, such as the crossing of a pedestrian or an obstacle on the highway. To obtain the theoretical mathematic expression, the braking of the PL is simplified as v 0 (t start ) = 0, where t start is the moment at which the PL stops immediately. It is reasonable to regard this as the most urgent case in braking. In addition, the string stability must be ensured so that the optimized parameters of PM1 can be extended to other following PMs [13], [30]. Thus, in the following description, d 0, and v 0 (t) = 0 if t ≥ t start , and s(t) = −u(t), i.e., s(t) represents deceleration and s(t) ≥ 0.
Considering the effect of delay, to reduce the standstill spacing and the duration of braking while ensuring safety and string stability, the optimization problem is modeled as follows: where d end is denoted as the standstill spacing, i.e., the final distance at the end of braking, t end is the entire duration of braking, d safe is the safe distance to avoid a rear-end collision, and s max is the safe permissible maximum value of deceleration to avoid damage to the vehicle structures and endangering passenger safety. In the optimization problem P, there are two optimization objectives for reducing VOLUME 8, 2020 the standstill spacing and the braking duration. Constraints (a) and (b) ensure inter-vehicle safety and in-vehicle safety, respectively. Constraints (c), (d), and (e) ensure the string stability of the platoon, where (a+2b)(d sparse −d dense )−2v max 2v max (a+b) = τ string is the maximum permissible delay to ensure the string stability of a platoon with the control law in Eq. (1) [6]. Thus, the optimization of P is based on the string stability of the platoon, which is guaranteed by constraints (c-e).
It is difficult to solve this optimization problem. On the one hand, due to the delay in the car-following interactions and the nonlinearity of the control law, there is no analysis model that can quantize the duration and the decreasing distance in the elapsed time of braking control. On the other hand, the optimization of these control parameters is an NP-hard problem, and a solution framework with low computational complexity must be provided. These two challenges are addressed in Sections IV and V.

IV. ANALYSIS OF STANDSTILL SPACING AND BRAKING DURATION BY DDEs
To analyze the final standstill spacing d end and braking duration t end , an overview analysis of two braking scenarios and the reason for their existence are presented in subsection IV-A. Then, subsection IV-B proposes a crossing criterion for judging to which scenario the braking process belongs. Then, in subsection IV-C, the time functions d(t) and v(t) are analyzed. Finally, the final standstill spacing d end and braking duration t end are derived in subsection IV-D.

A. ANALYSIS OF THE BRAKING PROCESS
Beginning at t start , PM1 will continue driving at v stable until t = t start +τ due to the existence of the delay τ , and this phase is defined as Delay phase. Then, PM1 starts to enter Braking phase, during which its starting moment is denoted as t brake , i.e., t brake = t start + τ . Here, t brake = 0 for the convenience of the analysis of the DDE in subsection IV-B; thus, t start = −τ . During Braking phase, v(t) and d(t) decrease continuously from v stable and d stable with t until v(t) = 0. In addition, the range of d stable can be derived as d dense ≤ d stable ≤ d sparse from Eqs. (1) and (2). Thus, there are two types of braking stages based on Eq. (2), and their control laws are as follows.
Braking stage 1: Braking stage 2: Based on whether v(t) has reduced to 0 in Braking stage 1, the braking process can be classified into Braking scenario 1 and Braking scenario 2.

1) BRAKING SCENARIO 1
As presented in Fig. 2, in Braking scenario 1, PM1 experiences Delay phase and Braking stage 1 and finally stops in Braking stage 1.
First, the expressions of d(t) and v(t) in Delay phase are as follows.
Then, in Braking phase, PM1 will brake with the control law of Braking stage 1. Thus, by substituting Eq. (2b) into Eq. (1), the expressions of u(t) and d(t) can be derived as follows. (8) It can be seen that Eq. (8) is a DDE with delay τ . The analysis of Eq. (8) is presented in subsection IV-C.

2) BRAKING SCENARIO 2
As presented in Fig. 3, in Braking scenario 2, PM1 experiences Delay phase, Braking stage 1, and Braking stage 2, and it finally stops in Braking stage 2.
In Braking scenario 2, as shown in Fig. 3, the red dot denotes that d(t) = d dense , and this moment is denoted as t dense . Due to the existence of delay τ , the control law will switch to Braking stage 2 at t wsitch = t dense + τ . As Delay phase and Braking stage 1 are the same as those in Braking scenario 1, Braking stage 2 is subsequently analyzed.
The equation of d(t) in Braking stage 2 is derived by substituting Eq. (2a) into Eq. (1), as follows: It can be seen that Eq. (9) is an ordinary differential equation (ODE) without any delay. Thus, the solution of Eq. (9) is The initial conditions of Eq. (10) are d(t wsitch ) = d wsitch and d (t wsitch ) = −v wsitch . It should be noted that the values of d wsitch and v wsitch can be obtained as the final values of Braking stage 1.
By substituting the initial conditions into Eq. (9), the expressions of d(t) and v(t) in Braking stage 2 are derived as: Based on the preceding analysis, the braking processes for both braking scenarios are modeled. In subsection IV-B, the crossing criterion for judging whether the braking process belongs to Braking scenario 1 or Braking scenario 2 is presented.

B. CROSSING CRITERION FOR DIFFERENTIATING BRAKING SCENARIOS
As revealed by the analysis of the braking process based on Figs. 2 and 3, d(t) = d dense is the condition for judging which braking scenario the braking process belongs to. The equivalence substitution is then conducted as follows: The judgment of d(t) = d dense can then be transformed into the judgment of r(t) = 0. By deriving two sides of Eq. (12a), the following can be obtained: Then, by substituting Eqs. (12b) and (13) into Eq. (5), the following can be obtained: Here, r(t) has the following form if the solution of Eq. (14) is nonoscillatory and decreasing [31]: where z is a positive number greater than 0. Substitute +d dense into Eq. (15) to obtain the value of A. Then, by substituting the value of A into Eq. (15) and Eq. (12a), the expressions of r(t) and d(t) are (17) If the solution to Eq. (14) is nonoscillatory and decreasing, r(t) will never cross r = 0, i.e., this braking process will belong to Braking scenario 1; otherwise, it will belong to Braking scenario 2. Thus, the following crossing criterion is defined. VOLUME 8, 2020 Crossing criterion: The braking process is Braking scenario 1, i.e., the solution of Eq. (13) is nonoscillatory and decreasing if the following inequality is satisfied: Proof: The following can be obtained by the first and second derivatives of Eq. (15): By substituting Eqs. (15), (19), and (20) into Eq. (14), Eq. (14) is transformed by simplification as follows: As a result, the crossing criterion is transformed into the existence of a positive z in Eq. (21) [31], [32].
However, Eq. (21) is a transcendental equation, and the analytic solution of z cannot be obtained; therefore, the proposition is transformed into the existence of a positive zero-point of f (z), where The first derivative of f (z) is Observably, f (0) > 0, f (0) < 0, and f (+∞) → +∞, and f (z) increases monotonically, i.e., f (z) only has one positive zero-point and f (z) has a minimum value for z > 0. Hence, the positive zero-point z 0 of f (z) is The conclusion can be drawn that the DDE (14) is non-crossing and decreasing, i.e., the braking process is Braking scenario 1, if Eqs. (24) and (25) are satisfied.
The crossing criterion comprises a, b, v max , d dense , and d sparse , and is not related to v stable ; in other words, a, b, v max , d dense , and d sparse determine whether the braking process belongs to Braking scenario 1 or 2.

C. TIME FUNCTIONS OF DISTANCE AND VELOCITY IN THE BRAKING PROCESS
In Sections IV-A and IV-B, it can be seen that there are two types of braking scenarios, namely Braking scenario 1 and Braking scenario 2, and both consist of Delay phase and Braking phase. The difference between the two braking scenarios is the element of Braking phase; there exists only Braking stage 1 for Braking scenario 1 while the additional Braking stage 2 is included for Braking scenario 2. The time function of distance (velocity) in Braking scenario 1 is a piecewise function of two segments, while that for Braking scenario 2 is a piecewise function of three segments, in which the former two segments are the same as those in Braking scenario 1. It should be noted that the value of the end-point of each segment is the initial value of the following segment. The time functions of all the segments are subsequently presented.

1) TIME FUNCTIONS OF DISTANCE AND VELOCITY IN DELAY PHASE
For compactness of expression and the benefit of substitution in the subsequent segments, d(t) and v(t) are equivalently transformed into the matrix form based on Eqs. (7a) and (7b):

2) TIME FUNCTIONS OF DISTANCE AND VELOCITY IN BRAKING STAGE 1
Based on Eqs. (5) and (8), the expressions of d (t) and v (t) in Braking stage 1 can be transformed into the following matrix form: where .
It can be seen that Eq. (27) is a set of DDEs, and the Runge-Kutta method with four orders (RK4) is introduced to obtain the approximate solution with high precision in this paper, the time step of which is denoted as t. It can be determined from Eq. (27) that as d(t) and v(t) are related to the state of τ prior to t in Braking stage 1, the values of d(t) and v(t) in Eq. (26) must be obtained at the steps in Delay phase.
Proposition 1: In Delay phase, the values of d(t) and v(t) of each point in t of PM1 can be obtained as follows: where t n denotes the n-th point of t, while y n , d n , and v n respectively represent the approximate values of y, d(t), and v(t) at t n . The time functions of d(t) and v(t) in Braking stage 1 can then be derived based on RK4 [33] by substitution in Eq. (27), which is represented as Proposition 2.
where k n,i denotes the i-th approximate growth rate of y n . In addition, k n,i = d (P(n t) + Q) /dt(n ≤ 0, i = 1, 2, 3) and the value of y n (n ≤ 0) is obtained by Proposition 1.
Proof: Please see the proof for Proposition 2 in the Appendix.
Proof: Please see the proof for Proposition 3 in the Appendix.

Proposition 4:
The values of d end and t end in Braking scenario 2 are obtained as follows: where t switch = min n:d n−τ/ t ≤d dense n t, Proof: Please see the proof for Proposition 4 in the Appendix.

V. THE RSD ALGORITHM: PSO BASED ON RESPONSE FOR DOUBLE OBJECTIVES AND CONSTRAINTS
As per the analysis of d end and t end presented in Section IV, the correspondences between these parameters (d end and t end ) and the optimization variables (a, b, d dense and d sparse ) in P are uniquely determined. To overcome the intractability of P, a solution framework with low complexity must be designed to obtain the approximate solution. PSO is a widely used optimization algorithm proposed by Kennedy and Eberhart [34], which can optimize multiple parameters simultaneously with a fast convergence rate and low complexity [13], [35]. Therefore, as P is an NP-hard problem comprising four optimization variables, PSO is utilized in this paper. However, there are two differences compared with traditional PSO problems. First, there are two objectives, rather than one, in P. Second, there are constraints existing in P. To address these differences, a particular fitness function must be properly designed. Thus, the lexicographic method [36] is introduced, as it is able to guarantee performance in a sequence of the importance of objectives. Additionally, a penalty function [37] is introduced to transform the constrained optimization problem P into an unconstrained optimization problem so that each particle can have a corresponding fitness value in PSO.
This section proposes a solution framework called the reduction of standstill spacing and braking duration (RSD) algorithm, which utilizes PSO combined with the lexicographic method and penalty function. The design of the fitness function in the proposed RSD algorithm to evaluate particles is introduced in subsection V-A. Then, subsection V-B presents the workflow of the proposed RSD algorithm.

A. FITNESS FUNCTION FOR DOUBLE OBJECTIVES AND CONSTRAINTS
A fitness function is designed to address the multiple objectives and constraints in P. First, x is set as the iteration variable, and fv is set as the fitness value in the PSO fitness function As there are four optimization variables in P, including a, b, d dense and d sparse the form of vector x is x = (a, b, d dense , d sparse ). VOLUME 8, 2020

1) THE LEXICOGRAPHIC METHOD FOR SEQUENCE OPTIMIZATION AND THE PENALTY FUNCTION FOR TRANSFORMING THE CONSTRAINTS
As P is a double-objective optimization problem the lexicographic method [36] is introduced by optimizing two single-objective subproblems in the sequence of importance. In this paper, compared with t end , d end is closely related to road utilization and safety insurance. Therefore, d end is taken as the first objective, and t end is the second objective. Then, P is optimized by solving two suboptimization problems in sequence, named Sub-P1 and Sub-P2. For the former Sub-P1, its objective is min d end , and its constraints correspond to the constraints (a-e) in P, where constraint (a) is transformed into d end ≥ d safe . This is because d(t) decreases monotonically and d end is the minimum value of d(t). Here, d opt represents the minimal objective value in optimizing Sub-P1. For the latter suboptimization problem Sub-P2, after the optimization of Sub-P1, t end is optimized. To guarantee the performance of the first objective d end in the subsequent optimization, the constraint d end ≤ (1 + rel)d opt is added in Sub-P2, where rel denotes a relaxation to guarantee that Sub-P2 is convex and solved efficiently and within a suitable time frame [36]. Thus, the constraints of Sub-P2 correspond to constraints (a-e), where constraint (a) is transformed into d safe ≤ d end ≤ (1+rel)d opt . Based on the preceding analysis, an independent variable flag is introduced as an indicator to indicate whether d end in Sub-P1 (flag = 1) or t end in Sub-P2 (flag = 2) is obtained.
As constraints exist in Sub-P1 and Sub-P2, the iterative process in PSO cannot be directly executed. Thus, the penalty function is introduced to impart a large penalty value λ to the particles that do not satisfy the constraints in Sub-P1 and Sub-P2 [37] because both Sub-P1 and Sub-P2 are minimization problems. This can transform the constrained P into an unconstrained optimization problem for the execution of PSO. In addition, combined with the subproblem indicator flag, the fitness function F(.) is represented as follows: if the constraints are not satisfied (32)

2) THE DESIGNED FITNESS FUNCTION
As PM1 stops in Braking stage 1 in Braking scenario 1, and it stops in Braking stage 2 in Braking scenario 2, the determination of d end and t end in Braking scenario 1 is based on Proposition 3, while that in Braking scenario 2 is based on Proposition 4 by obtaining d switch and v switch based on Proposition 2. Then, based on the above analysis, the determination of fv in Braking scenarios 1 and 2 is designed in the rest of this part. First, the stability condition is judged for each particle by constraints (c-e) in P. Next, for stable particles, the crossing criterion is introduced to obtain fv in Braking scenarios 1 and 2, respectively.
Next, for the in-vehicle safety, the judgment corresponds to constraint (b) in Sub-P1 and Sub-P2, i.e., s(t) ≤ s max .
Because v n can be obtained by the introduced RK4, the deceleration at t n can be approximated and judged as where s n represents the approximate value of s(t n ).
Here, t n should be from −τ to t max , t max is an input denoted as the maximum permissible time for braking, and L = t max / t . For inter-vehicle safety, constraint (a) in Sub-P1 and Sub-P2 must be judged as follows: If the string stability is ensured, the in-vehicle and inter-vehicle safety are satisfied, and we have fv = d end , if flag = 1, and fv = t end , if flag = 2. Otherwise, if any one of the constraints is not satisfied, fv = λ.
The determination of the fitness function is described in Algorithm 1.

B. THE PROPOSED RSD ALGORITHM
The proposed RSD algorithm based on PSO is subsequently presented in detail. The symbols used in PSO are listed in Table 1. The procedures of the proposed RSD algorithm are then presented. Here, the complexity of the proposed RSD algorithm is polynomial O(LNK ). Please see the proof of the complexity computation in the Appendix.

VI. SIMULATION RESULTS
This section consists of three parts. Subsection VI-A demonstrates the accuracy of the analysis of standstill spacing and braking duration by the DDEs, including the crossing criterion and the determination of d end and t end based on RK4. Next, in subsection VI-B, the improved performance of the proposed RSD algorithm is presented via comparison with four related algorithms, namely, the car-following algorithm [25], the feedback control algorithm [26], the maximum braking algorithm [27], and the fixed setting algorithm using Algorithm 1 Construction of the Fitness Function fv = F(x, λ, flag) Inputs: x, λ, flag, v max , v stable , d safe , s max , ε, t, τ, rel, L Output: fv = F(x, λ, flag) 1: Initialize fv = λ 2: if the platoon is string stable based on constraints (c-e) in P // ensure string stability then 3: Obtain d stable based on Eq. (3) 4: Obtain d n and v n for n = −τ / t, . . . , 0 based on Proposition 1 5: for each t n , n = 1, 2, 3, . . . , L do 6: Obtain d n and v n based on Proposition 2 7: if Inequality (33) is satisfied // ensure in-vehicle safety then 8: if the braking process belongs to Braking scenario 1 based on crossing criterion then 9: if v n ≤ ε then 10: d end = d dense and t end = n t 11:

Goto
Step 23 12: end if 13: else 14: if d n−τ / t < d dense then 15: Obtain d end and t end based on Proposition 4.

Goto
Step 26 21: end if 22: end for 23: if Inequality (34) is satisfied // ensure inter-vehicle safety then 24: Obtain fv based on flag 25: end if 26: end if the same control laws [6]. Finally, in subsection VI-C, the relationship among the limits of the stable velocity, the delay and the stable distance is presented as a reference for platoon control while ensuring safety.

A. THE ACCURACY OF MODELING THE BRAKING PROCESS 1) CROSSING CRITERION
A crossing criterion defined in Section IV-B is proposed to divide the braking process into Braking scenario 1 and Braking scenario 2, and its accuracy is subsequently verified Fig. 4 presents the actual results of the classification of braking scenarios with different delays and stable velocities, in which it can be seen that the percent of braking in Braking scenario 1 is greater with shorter delays and larger values of v stable . In addition, the braking process tends to be Braking scenario 1 when the delays are close to 0. This is Algorithm 2 The Proposed RSD Algorithm Inputs: N , K , x min , x max , v min , v max , c 1 , c 2 , ω.
1: Conduct PSO where the fitness function is fv = F(x, λ, 1) and the optimization objective is min d end . 2: After PSO in step 1, it can be obtained that d opt = F(x gbest , λ, 1) 3: Conduct PSO where the fitness function is fv = F(x, λ, 2) and the optimization objective is min t end .
Here, d opt is introduced in this step because constraint (a) in fv = F(x, λ, 2) is d safe ≤ d end ≤ (1 + rel)d opt to illuminate particles that cannot guarantee the performance of the first objective d end in step 1. Outputs: d end = F(x gbest , λ, 1), t end = F(x gbest , λ, 2), results of optimization variables x gbest . because when delays are close to 0, the DDE (8) of Braking stage 1 tends to be an ODE, the solution of which is nonoscillatory and decreasing, i.e., the braking process belongs to Braking scenario 1 when τ = 0.
As shown in Fig. 5, PM1 has a longer d stable to brake when it drives with a higher value of v stable , and the braking process tends to be Braking scenario 1. Therefore, a shorter time delay and larger d stable make the braking process more relaxed. Fig. 6 presents the accuracy of the crossing criterion with different delays and stable velocities. For each point in Fig. 6, the optimization variables including a, b, d dense , and d sparse , are optimized by the proposed RSD algorithm. The accuracy of the crossing criterion is found to be 94.02%. The distribution of inaccurate points of the crossing criterion is close to a line (represented by blue dots) because the value of f (z 0 ) corresponding to this line is close to 0, leading to the inaccuracy of the crossing criterion.
2) THE DETERMINATION OF d end AND t end Fig. 7 presents the percentage of differences between the theoretical and simulated values for d end and t end . As presented in Fig. 7(a), the percentage of differences for d end is less than VOLUME 8, 2020  0.18%, while that for t end is less than 0.03%, as presented in Fig. 7(b). Thus, the propositions for the determination of d end and t end are highly accurate.

B. THE PERFORMANCE OF THE PROPOSED RSD ALGORITHM
To evaluate the performance of the proposed RSD algorithm, it is compared with four related algorithms, namely, the car-following algorithm [25], the feedback control algorithm [26] the maximum braking algorithm [27], and the fixed setting algorithm, which has the same control laws as the RSD algorithm. The parameters are listed in Table 2, and the parameters of the fixed setting algorithm are set as a = 4, b = 0.6, d sparse = 35 m, and d dense = 1.5d safe /d dense = 2d safe [6].  PM1 starts to brake at t brake ; hence, s(0) can be derived as 202454 VOLUME 8, 2020 For the constraint s(t) ≤ s max , it should be ensured that bv stable ≤ s max . Therefore, b ≤ u max /v stable , and the maximum deceleration value for a vehicle is 10 m/s 2 [38].
The maximum value of d sparse is 100 m because the maximum safe distance between vehicles for a platoon requires d stable to be 100 m [39] when it drives at a speed of 30 m/s. In addition, min d sparse ≥ max d dense , or the control law (2) will not work if d sparse < d dense .
Furthermore, the control gains a and b must be guaranteed as a > 0 and b > 0 so that min a = min b = 0. Fig. 8 illustrates the standstill spacing and braking duration under different delays, which represent inter-vehicle safety ( Fig. 8(a)) and road utilization (Figs. 8(a) and 8(b)). It is evident that the fixed setting algorithm and the feedback control algorithm cannot guarantee inter-vehicle safety, i.e., d end < d safe and the unsafe results are represented as hollow points. For the fixed setting algorithm, it is found that d end decreases with the increase of delays, and it finally induces collision with a high delay (τ > 0.5 s for d dense = 1.5d safe , τ > 0.6 s for d dense = 2d safe ) because the fixed control parameters in control laws (1) and (2) cannot cope with the worsened delay condition. The feedback control algorithm lacks both the offset of delay in distance estimation and adequate deceleration. Fig. 9 demonstrates the distribution of deceleration under different delays and control input dependent on time when τ = 0.4 s, which represents in-vehicle safety. It is noted that deceleration is equal to the absolute value of control input, so that deceleration is greater than 0 and control input is less than 0. It is clear that the car-following algorithm cannot guarantee in-vehicle safety, i.e., s > s max and the unsafe results are represented as hollow columns and dotted lines. This is because the insurance of in-vehicle safety is not considered in the car-following algorithm, which uses an overly large deceleration in the decelerating phase. In addition, Fig. 8(b) shows that the braking duration in the proposed RSD algorithm is less than 4 s, while that in the car-following algorithm ranges from 5.2 s to 7 s. In other words, it has a longer braking duration compared with the proposed algorithm, even though it exceeds the limit of deceleration in the decelerating phase.
It is evident in Figs. 8 and 9 that, aside from the proposed RSD algorithm, only the maximum braking algorithm can ensure safety, and it is found to have a shorter braking duration based on Fig. 8(b). However, the standstill spacing in the maximum braking algorithm is greater than 8 m based on Fig. 8(a), which is not beneficial for improving road utilization. Moreover, under this algorithm, the vehicles will continuously brake at the maximum deceleration based on Fig. 9(a), and the control input will experience a sudden fall and rise at the beginning and end of the braking process based on Fig. 9(b), which will have a sharp and substantial impact on passengers, who will be made to feel extremely uncomfortable. In summary, only the proposed RSD algorithm can simultaneously improve road utilization and ensure the two aspects of safety.

C. RESULTS FOR REFERENCE
The relationship among the limits of the stable velocity, the delay, and the stable distance is presented for reference in platoon control while ensuring safety. Fig. 10 presents the minimum stable distance for ensuring safety at different delays and stable velocities. If the stable distance is shorter than the minimum value, the standstill spacing may be less than the safe distance, or the deceleration may be beyond the permissible value. In Fig. 10, it can be seen that the minimum stable distance increases with the stable velocity and delay because a longer distance is required to safely brake when vehicles drive at higher speeds or under longer delays.

VII. CONCLUSION
This paper proposes an optimized braking control under delays in which road utilization is improved by reducing the standstill spacing and braking duration while ensuring safety. In particular, the braking process is modeled using DDEs to embed the effect of delay. In addition, a crossing criterion is introduced to classify the braking scenarios. Furthermore, the time functions of the distance and velocity during braking are derived, and propositions of the standstill spacing and braking duration are made based on the Runge-Kutta method. Then, under the consideration of ensuring safety with delays, the road utilization is improved via the optimization of the control parameters, and a solution framework with polynomial complexity O(LNK ) is provided. The simulation results verify the accuracy of the braking modeling process, demonstrate the performance of the proposed algorithm, and provide results as a reference for platoon control.
Furthermore, braking with time-variant delays, heterogeneous platoons and different information topologies can be studied in the future. Moreover, the lateral behavior of a platoon during the braking process will be considered in the future

APPENDIX A PROOF OF PROPOSITION 2
Proof: The initial value problem for the DDE (28) is as follows: where y(t) = Pt + Q is the initial function of the DDE.
Here, (t n , y n ) denotes the n-th value of (t, y), and this is required to obtain the value of (t n , y n ) for t n > 0, where t = 0 represents the initial moment of the equation.
The formula of the approximate solution of the DDE is as follows [40]: where f ni denotes the i-th approximate growth rate of (t n , y n ), h i is the weight of f ni , and s represents the order of the method, n = 0, 1, 2, . . .. Then f ni can be derived as follows [40]. a ij f nj (38) Note that S(t ni − τ ) denotes the approximate, not the exact, value of y(t ni −τ ). Thus, S(t ni −τ ) is different from y(t ni −τ ). Because t is a step factor of τ in the proposed algorithm, i.e., q = τ/ t, and q is an integer, (t ni − τ, y(t ni − τ )) is approximated by (t ni−q , y ni−q ), n = 0, 1, 2, . . .. Hence, S(t ni − τ ) is equal to y ni−q .
Based on the preceding analysis, the formula of RK4 can be obtained as Proposition 2 [40].
It is noted that y n = P(n t) + Q and k n,i = d(P(n t) + Q)/dt when n ≤ 0, i = 1, 2, 3, i.e., y n and k n,i are exact values when n ≤ 0 because the initial function is given, in which k n,i (n ≤ 0, i = 1, 2, 3) are equal.

APPENDIX B PROOF OF PROPOSITION 3
Proof: In Braking scenario 1, the forms of d(t) and v(t) are obtained as follows based on Eqs. (12a) and (15): i.e., d(t) will decrease to d dense , v(t) will decrease to 0, and d end = d dense .
Based on the analysis of Braking scenario 1 in Section IV-C, the braking process ends if v(t) ≤ ε. Therefore, t end = n t when it enters v n ≤ ε, i.e., t end = min n:v n ≤ε n t, because v(t) decreases monotonously with time, i.e., v n > ε if n < t end / t and v n ≤ ε if n ≥ t end / t.

APPENDIX C PROOF OF PROPOSITION 4
Proof: When t ≥ t switch , the equations of d(t) and v(t) are (11a) and (11b), respectively. Thus, d(t) and v(t) will be uniquely identified after t switch if t switch , d switch , and v switch are given.
Based on the analysis presented in Section IV-A, t dense is the moment at which d(t) = d dense , i.e., d(t) > d dense if t < t dense and d(t) < d dense if t > t dense . In other words, d(t) > d dense if t < t switch − τ and d(t) < d dense if t > t switch − τ , as t dense = t switch − τ . Thus, t switch = min n:d n−τ/ t ≤d dense n t according to the proof of Proposition 3. In addition, v switch = v n and d switch = d n at moment t = t switch , i.e., n = t switch / t. Therefore, t switch , d switch , and v switch are obtained in Braking stage 1 through Proposition 2.
Here, d(t) in Braking stage 2 decreases to d switch − v switch a+b so that d end = d switch − v switch a+b according to the proof of Proposition 3.
Because PM1 is considered to stop if v(t) ≤ ε, there are two cases for the determination of t end , as v(t) ≤ ε will be in Braking stage 1 or 2.
Case 1: v(t) ≤ ε is in Braking stage 1. In this case, v switch < ε because v switch is the final velocity of Braking stage 1. Thus, t end = min n:v n ≤ε n t based on the proof of Proposition 3. Case 2: v(t) ≤ ε is in Braking stage 2. In this case, v switch ≥ ε because v switch is the initial velocity of Braking stage 2. Because v(t) is decreasing in Braking stage 2, t end is obtained by solving the following equation: v switch e (a+b)(t switch −t end ) = ε (40) so that

APPENDIX D PROOF OF COMPLEXITY COMPUTATION
When computing the complexity, the order of magnitude of the most executed statements is equal to the complexity. In the RSD algorithm, the innermost loop is the obtaining of d end or t end in the fitness function, which should be executed at most L times. Then, in the PSO process, N particles will be updated K times, and one update contains the obtaining of d end or t end at most L times. Based on the above analysis, d end or t end will be obtained at most LNK times; namely, the complexity is O(LNK ), and it is polynomial complexity.